Problems arising from Balogh's “Locally nice spaces under Martin's Axiom”

Abstract

We discuss recent advances extending the paper mentioned in the title, in particular, the consistency proof for "all locally compact perfectly normal spaces are paracompact". As well, we point out mistakes in the published literature concerning metrizability of hereditarily normal manifolds, and set out an approach toward proving the consistency of such metrizability for those of dimension greater than 1.  2004 Elsevier B.V. All rights reserved.MSC: primary 54D45, 54D20, 54D15, 03E35, 54A35; secondary 03E35, 03E50, 54E35, 57N99

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

“…In [24], in a sketch of a plan to prove that PFA(S)[S] implies locally compact, hereditarily normal spaces not including a perfect pre-image of ω 1 are paracompact (which result we accomplished in a model of PFA(S) [S] in [15]), the author introduced:…”

Hereditarily normal manifolds of dimension > 1 may all be metrizable

“…The clever topological ideas of the proof are taken from [13, p189]. A sketch of this proof appears in [22]. The main idea of the proof is to use PPI + to find copies of ω 1 and, combined with Lemma 1.8, to show that, in fact, there are copies of the Tychonoff plank in the space.…”